Reduced Weyl asymptotics for pseudodifferential operators on bounded domains I. The finite group case

Let G ⊂ O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn, and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A0 in L2(Rn) with G-invariant Weyl symbol, and assume that it is semi-bounded from below. We show that the spectrum of the Friedrichs extension A of the operator res ◦ A0 ◦ ext : C∞c (X)→L2(X) is discrete, and derive asymptotics for the number Nχ (λ) of eigenvalues of A less or equal λ and with eigenfunctions in the χ-isotypic component of L2(X) as λ→∞, giving also an estimate for the remainder term in case that G is a finite group. In particular, we show that the multiplicity of each unitary irreducible representation in L2(X) is asymptotically proportional to its dimension. © 2008 Elsevier Inc. All rights reserved.